(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     14054,        449]
NotebookOptionsPosition[     12157,        384]
NotebookOutlinePosition[     12575,        400]
CellTagsIndexPosition[     12532,        397]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Daubechies Filters", "Title",
 CellChangeTimes->{{3.4217479401079364`*^9, 3.4217479429048653`*^9}}],

Cell["\<\
Wavelet Workshop
June 2-5, 2009
University of St. Thomas\
\>", "Subtitle",
 CellChangeTimes->{{3.421426484890625*^9, 3.42142648821875*^9}, {
  3.453030366875*^9, 3.4530303721875*^9}}],

Cell["\<\
Catherine Beneteau
Caroline Haddad
David Ruch
Patrick Van Fleet\
\>", "Subsubtitle",
 CellChangeTimes->{{3.42144927371875*^9, 3.42144928703125*^9}}],

Cell[CellGroupData[{

Cell["Objective", "Section",
 CellChangeTimes->{{3.4217479766243973`*^9, 3.421747978296315*^9}}],

Cell["\<\
The purpose of this notebook is to introduce you to Daubechies filters and \
see how they can be used in applications.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["DiscreteWavelets", "Section"],

Cell[TextData[{
 StyleBox["You should run this cell each time you open this notebook!!",
  FontColor->RGBColor[1, 0, 0]],
 "  It loads the ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " package ",
 StyleBox["DiscreteWavelets",
  FontFamily->"Courier"],
 " for use in subsequent computations."
}], "Text"],

Cell[BoxData[
 RowBox[{"<<", "DiscreteWavelets`DiscreteWavelets`"}]], "Input",
 CellChangeTimes->{{3.421749074496252*^9, 3.421749074590005*^9}, {
  3.453030844328125*^9, 3.453030845421875*^9}},
 FontColor->GrayLevel[0.500008]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Help on DiscreteWavelets", "Section"],

Cell[TextData[{
 "If you ever need help with ",
 StyleBox["DiscreteWavelets",
  FontFamily->"Courier"],
 ", go to the ",
 StyleBox["Documentation Center",
  FontSlant->"Italic"],
 ", then look at the lower right for a link called ",
 StyleBox["Installed Add-Ons",
  FontSlant->"Italic"],
 ".  Click this and then click ",
 StyleBox["DiscreteWavelets",
  FontSlant->"Italic"],
 ". "
}], "Text",
 CellChangeTimes->{{3.421748017312939*^9, 3.4217480658454313`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solving for the D4 Filter", "Section",
 CellChangeTimes->{{3.4217480856584387`*^9, 3.421748091408586*^9}}],

Cell["We first set up the orthogonality conditions", "Text",
 CellChangeTimes->{{3.421748098877527*^9, 3.4217481194249277`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Clear", "[", 
   RowBox[{"h", ",", "h0", ",", "h1", ",", "h2", ",", "h3"}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"h", "=", 
   RowBox[{"{", 
    RowBox[{"h0", ",", "h1", ",", "h2", ",", "h3"}], "}"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{"orth", "=", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{
     RowBox[{"Total", "[", 
      RowBox[{"h", "^", "2"}], "]"}], "\[Equal]", "1"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"h0", "*", "h2"}], "+", 
      RowBox[{"h1", "*", "h3"}]}], "\[Equal]", "0"}]}], "}"}]}]}], "Input",
 CellChangeTimes->{{3.421748125471958*^9, 3.42174818355157*^9}, {
  3.4217489035856276`*^9, 3.4217489036950045`*^9}, {3.4217490726680803`*^9, 
  3.4217490727930837`*^9}, {3.453030848203125*^9, 3.45303084896875*^9}}],

Cell["Next the lowpass conditions ", "Text",
 CellChangeTimes->{{3.42174818494223*^9, 3.421748239146743*^9}, {
  3.45303038496875*^9, 3.453030385125*^9}}],

Cell[BoxData[
 RowBox[{"lowpass", "=", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{
     RowBox[{"Total", "[", "h", "]"}], "\[Equal]", 
     RowBox[{"Sqrt", "[", "2", "]"}]}], ",", 
    RowBox[{
     RowBox[{"h", ".", 
      RowBox[{"{", 
       RowBox[{"1", ",", 
        RowBox[{"-", "1"}], ",", "1", ",", 
        RowBox[{"-", "1"}]}], "}"}]}], "\[Equal]", "0"}], ",", 
    RowBox[{
     RowBox[{"h", ".", 
      RowBox[{"{", 
       RowBox[{"0", ",", 
        RowBox[{"-", "1"}], ",", "2", ",", 
        RowBox[{"-", "3"}]}], "}"}]}], "\[Equal]", "0"}]}], "}"}]}]], "Input",\

 CellChangeTimes->{{3.4217482055365076`*^9, 3.421748258475363*^9}, {
  3.421748904804408*^9, 3.421748904945037*^9}, {3.4217490716211786`*^9, 
  3.421749071746182*^9}, {3.453030391296875*^9, 3.45303043871875*^9}, {
  3.4530308516875*^9, 3.45303085234375*^9}}],

Cell["Now we solve the system", "Text",
 CellChangeTimes->{{3.421748276100814*^9, 3.421748281772834*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"eqs", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"orth", ",", "lowpass"}], "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"s", "=", 
  RowBox[{"Solve", "[", 
   RowBox[{"eqs", ",", "h"}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  RowBox[{"\"\<There are \>\"", ",", 
   RowBox[{"Length", "[", "s", "]"}], ",", "\"\< solutions.\>\""}], 
  "]"}]}], "Input",
 CellChangeTimes->{{3.421748291179325*^9, 3.4217484861530657`*^9}, {
  3.4217489021637154`*^9, 3.421748902304344*^9}, {3.4217490693398705`*^9, 
  3.421749069480499*^9}, {3.453030465203125*^9, 3.453030476078125*^9}, {
  3.453030855234375*^9, 3.453030856234375*^9}}],

Cell[TextData[{
 "Parsing out the solutions from a Solve call is a bit cumbersome in ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ".  Here is some code to do it."
}], "Text",
 CellChangeTimes->{{3.453030483078125*^9, 3.453030508015625*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"sln1", "=", 
   RowBox[{"h", "/.", 
    RowBox[{"s", "[", 
     RowBox[{"[", "1", "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"sln2", "=", 
   RowBox[{"h", "/.", 
    RowBox[{"s", "[", 
     RowBox[{"[", "2", "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{"\"\<The first solution is \>\"", ",", 
    RowBox[{"TableForm", "[", "sln1", "]"}], ",", "\"\<.\\n\>\""}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"Print", "[", 
    RowBox[{"\"\<The second solution is \>\"", ",", 
     RowBox[{"TableForm", "[", "sln2", "]"}], ",", "\"\<.\>\""}], "]"}], 
   ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", "\"\<Numerical versions:\\n\>\"", "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{"\"\<The first solution is \>\"", ",", 
    RowBox[{"TableForm", "[", 
     RowBox[{"N", "[", "sln1", "]"}], "]"}], ",", "\"\<.\\n\>\""}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{"\"\<The second solution is \>\"", ",", 
    RowBox[{"TableForm", "[", 
     RowBox[{"N", "[", "sln2", "]"}], "]"}], ",", "\"\<.\>\""}], "]"}], 
  ";"}]}], "Input",
 CellChangeTimes->{{3.421748291179325*^9, 3.4217484861530657`*^9}, {
   3.4217489021637154`*^9, 3.421748902304344*^9}, {3.4217490693398705`*^9, 
   3.421749069480499*^9}, 3.453030465203125*^9, {3.453030858453125*^9, 
   3.45303085909375*^9}}],

Cell["Note that the second solution is a reflection of the first.", "Text",
 CellChangeTimes->{{3.421748489090641*^9, 3.42174849782524*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Highpass Filter", "Section",
 CellChangeTimes->{{3.42174876719151*^9, 3.4217487693634415`*^9}}],

Cell["\<\
We obtain the highpass filter by reversing the solution and multiplying each \
coefficient alternately  by 1 and - 1.\
\>", "Text",
 CellChangeTimes->{{3.421748777098014*^9, 3.421748813177063*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"lp", "=", "sln2"}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"hp", "=", 
   RowBox[{
    RowBox[{"Reverse", "[", "lp", "]"}], "*", 
    RowBox[{"{", 
     RowBox[{"1", ",", 
      RowBox[{"-", "1"}], ",", "1", ",", 
      RowBox[{"-", "1"}]}], "}"}]}]}], ";"}]}], "Input",
 CellChangeTimes->{{3.421748818786581*^9, 3.4217488986948767`*^9}, {
  3.421749065777279*^9, 3.4217490659491587`*^9}, {3.45303053371875*^9, 
  3.453030556703125*^9}, {3.453030668234375*^9, 3.45303067296875*^9}, {
  3.45303072209375*^9, 3.453030749390625*^9}, {3.453030861828125*^9, 
  3.4530308623125*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Wavelet Matrix", "Section",
 CellChangeTimes->{{3.421748660438778*^9, 3.4217486630169687`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"n", "=", "8"}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"r1", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"lp", ",", 
     RowBox[{"Table", "[", 
      RowBox[{"0", ",", 
       RowBox[{"{", 
        RowBox[{"n", "-", "4"}], "}"}]}], "]"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"top", "=", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"RotateRight", "[", 
      RowBox[{"r1", ",", 
       RowBox[{"2", "*", "k"}]}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"k", ",", "0", ",", 
       RowBox[{
        RowBox[{"n", "/", "2"}], "-", "1"}]}], "}"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"r2", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"hp", ",", 
     RowBox[{"Table", "[", 
      RowBox[{"0", ",", 
       RowBox[{"{", 
        RowBox[{"n", "-", "4"}], "}"}]}], "]"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"bot", "=", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"RotateRight", "[", 
      RowBox[{"r2", ",", 
       RowBox[{"2", "*", "k"}]}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"k", ",", "0", ",", 
       RowBox[{
        RowBox[{"n", "/", "2"}], "-", "1"}]}], "}"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"W", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"top", ",", "bot"}], "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{
   "\"\<The \>\"", ",", "n", ",", "\"\<x\>\"", ",", "n", ",", 
    "\"\< wavelet matrix is \>\"", ",", 
    RowBox[{"MatrixForm", "[", "W", "]"}], ",", "\"\<.\>\""}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{"\"\<The product of W and its transpose is \>\"", ",", 
    RowBox[{"MatrixForm", "[", 
     RowBox[{"Simplify", "[", 
      RowBox[{"W", ".", 
       RowBox[{"Transpose", "[", "W", "]"}]}], "]"}], "]"}], ",", 
    "\"\<.\>\""}], "]"}], ";"}]}], "Input",
 CellChangeTimes->{{3.4217486744235106`*^9, 3.421748763066405*^9}, {
  3.421748896147937*^9, 3.4217488965229464`*^9}, {3.4217490393547277`*^9, 
  3.421749064011609*^9}, {3.453030565046875*^9, 3.45303061975*^9}, {
  3.453030650203125*^9, 3.453030662828125*^9}, {3.45303075971875*^9, 
  3.453030816484375*^9}, {3.453030865484375*^9, 3.453030865921875*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Daubechies Filters in DiscreteWavelets", "Section",
 CellChangeTimes->{{3.4217489075076027`*^9, 3.4217489150859213`*^9}}],

Cell[TextData[{
 "We can generate Daubechies filters in DiscreteWavelets using the ",
 StyleBox["Daub",
  FontColor->RGBColor[1, 0, 0]],
 " command."
}], "Text",
 CellChangeTimes->{{3.421748919429783*^9, 3.421748933633271*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.4217489493367987`*^9, 3.4217489510712175`*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "4", "]"}]], "Input",
 CellChangeTimes->{{3.421748953165021*^9, 3.4217489539150405`*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "6", "]"}]], "Input",
 CellChangeTimes->{{3.4217489573526287`*^9, 3.4217489590245466`*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "12", "]"}]], "Input",
 CellChangeTimes->{{3.4217489615089846`*^9, 3.4217489635715375`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Exercise", "Section",
 CellChangeTimes->{{3.421748966024726*^9, 3.421748967524764*^9}}],

Cell["Can you write code to find the D6 filter? The D8 filter?", "Text",
 CellChangeTimes->{{3.4217489728530254`*^9, 3.4217489812282395`*^9}, {
  3.4530307345625*^9, 3.453030737546875*^9}}],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{"Put", " ", "code", " ", "here"}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.42174898672838*^9, 3.421748990009714*^9}}]
}, Open  ]]
}, Open  ]]
},
WindowSize->{1272, 683},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (April 20, 2007)",
StyleDefinitions->FrontEnd`FileName[{"Creative"}, "NaturalColor.nb", 
  CharacterEncoding -> "WindowsANSI"]
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 105, 1, 73, "Title"],
Cell[698, 26, 193, 6, 96, "Subtitle"],
Cell[894, 34, 158, 6, 89, "Subsubtitle"],
Cell[CellGroupData[{
Cell[1077, 44, 96, 1, 75, "Section"],
Cell[1176, 47, 142, 3, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1355, 55, 35, 0, 75, "Section"],
Cell[1393, 57, 312, 10, 29, "Text"],
Cell[1708, 69, 226, 4, 41, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1971, 78, 43, 0, 75, "Section"],
Cell[2017, 80, 461, 15, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2515, 100, 112, 1, 75, "Section"],
Cell[2630, 103, 128, 1, 29, "Text"],
Cell[2761, 106, 809, 22, 82, "Input"],
Cell[3573, 130, 154, 2, 29, "Text"],
Cell[3730, 134, 836, 23, 41, "Input"],
Cell[4569, 159, 105, 1, 29, "Text"],
Cell[4677, 162, 674, 15, 82, "Input"],
Cell[5354, 179, 248, 6, 29, "Text"],
Cell[5605, 187, 1535, 40, 182, "Input"],
Cell[7143, 229, 140, 1, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[7320, 235, 105, 1, 75, "Section"],
Cell[7428, 238, 207, 4, 29, "Text"],
Cell[7638, 244, 625, 15, 62, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[8300, 264, 105, 1, 75, "Section"],
Cell[8408, 267, 2316, 68, 182, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[10761, 340, 127, 1, 75, "Section"],
Cell[10891, 343, 227, 6, 29, "Text"],
Cell[11121, 351, 127, 2, 41, "Input"],
Cell[11251, 355, 125, 2, 41, "Input"],
Cell[11379, 359, 127, 2, 41, "Input"],
Cell[11509, 363, 128, 2, 41, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[11674, 370, 93, 1, 75, "Section"],
Cell[11770, 373, 189, 2, 29, "Text"],
Cell[11962, 377, 167, 3, 41, "Input"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)